3-3 Complex Numbers. Simplify. SOLUTION: 2. SOLUTION: 3. (4i)( 3i) SOLUTION: 4. SOLUTION: 5. SOLUTION: esolutions Manual - Powered by Cognero Page 1
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1 1. Simplify (4i)( 3i) esolutions Manual - Powered by Cognero Page 1
2 6. 7. Solve each equation. 8. Find the values of a and b that make each equation true. 9. 3a + (4b + 2)i = 9 6i Set the real parts equal to each other. Set the imaginary parts equal to each other. esolutions Manual - Powered by Cognero Page 2
3 10. 4b 5 + ( a 3)i = 7 8i Set the real parts equal to each other. Set the imaginary parts equal to each other. Simplify. 11. ( 1 + 5i) + ( 2 3i) 12. (7 + 4i) (1 + 2i) 13. (6 8i)(9 + 2i) 14. (3 + 2i)( 2 + 4i) esolutions Manual - Powered by Cognero Page 3
4 esolutions Manual - Powered by Cognero Page 4
5 17. ELECTRICITY The current in one part of a series circuit is 5 3j amps. The current in another part of the circuit is 7 + 9j amps. Add these complex numbers to find the total current in the circuit. 18. Simplify ( 3i)( 7i)(2i) esolutions Manual - Powered by Cognero Page 5
6 23. 4i( 6i) i i (10 7i) + (6 + 9i) 27. ( 3 + i) + ( 4 i) 28. (12 + 5i) (9 2i) 29. (11 8i) (2 8i) esolutions Manual - Powered by Cognero Page 6
7 30. (1 + 2i)(1 2i) 31. (3 + 5i)(5 3i) 32. (4 i)(6 6i) 33. esolutions Manual - Powered by Cognero Page 7
8 Solve each equation. esolutions Manual - Powered by Cognero Page 8
9 esolutions Manual - Powered by Cognero Page 9
10 41. Find the values of x and y that make each equation true i = 3x + 4yi Set the real parts equal to each other. Set the imaginary parts equal to each other. 43. x yi = 3 6i Set the real parts equal to each other. Set the imaginary parts equal to each other x (3 y)i = 4 + 6i Set the real parts equal to each other. 2x + 7 = 4 2x = 4 7 2x = 11 Set the imaginary parts equal to each other. 3 y = 6 y = 3 esolutions Manual - Powered by Cognero Page 10
11 y + (3x 7)i = 9 3i Set the real parts equal to each other. Set the imaginary parts equal to each other. 46. a + 3b + (3a b)i = 6 + 6i Set the real parts equal to each other. Set the imaginary parts equal to each other. Multiply the second equation by 3 and add the resulting equation to (1). Substitute in (1). esolutions Manual - Powered by Cognero Page 11
12 47. (2a 4b)i + a + 5b = i Set the real parts equal to each other. Set the imaginary parts equal to each other. Multiply the first equation by 2 and subtract the second equation from the resulting equation. Substitute in (1). 48. Simplify i 41 esolutions Manual - Powered by Cognero Page 12
13 51. (4 6i) + (4 + 6i) (4 6i) + (4 + 6i) = i + 6i = (8 5i) (7 + i) (8 5i) (7 + i) = 8 5i 7 i = 1 6i 53. ( 6 i)(3 3i) 54. esolutions Manual - Powered by Cognero Page 13
14 ( 4 + 6i)(2 i)(3 + 7i) 57. (1 + i)(2 + 3i)(4 3i) esolutions Manual - Powered by Cognero Page 14
15 ELECTRICITY The impedance in one part of a series circuit is 7 + 8j ohms, and the impedance in another part of the circuit is 13 4j ohms. Add these complex numbers to find the total impedance in the circuit. Total impedance = 7 + 8j j = j ohms esolutions Manual - Powered by Cognero Page 15
16 ELECTRICITY Use the formula 61. The current in a circuit is 3 + 6j amps, and the impedance is 5 j ohms. What is the voltage? We know that voltage can be calculated by. V = Voltage C = current I = impedance Therefore, the voltage is Volts. 62. The voltage in a circuit is 20 12j volts, and the impedance is 6 4j ohms. What is the current? We know that voltage can be calculated by. V = Voltage C = current I = impedance Therefore, the current is amps. 63. Find the sum of ix 2 (4 + 5i)x + 7 and 3x 2 + (2 + 6i)x 8i. ix 2 (4 + 5i)x x 2 + (2 + 6i)x 8i = (3 + i)x 2 5ix 4x + 2x + 6ix + 7 8i = (3 + i)x 2 + ix 2x + 7 8i = (3 + i)x 2 + ( 2 + i)x + 7 8i esolutions Manual - Powered by Cognero Page 16
17 64. Simplify [(2 + i)x 2 ix i] [( 3 + 4i)x 2 + (5 5i)x 6]. [(2 + i)x 2 ix i] [( 3 + 4i)x 2 + (5 5i)x 6] = [(2 + i)x 2 ix i] ( 3 + 4i)x 2 (5 5i)x + 6 = 2x 2 + ix 2 ix i + 3x 2 4ix 2 5x + 5ix + 6 = 5x 2 3ix 2 + i 5x + 4ix + 11 = (5 3i)x 2 + ( 5 + 4i)x + i MULTIPLE REPRESENTATIONS In this problem, you will explore quadratic equations that have complex roots. a. Algebraic Write a quadratic equation in standard form with 3i and 3i as its roots. b. Graphical Graph the quadratic equation found in part a by graphing its related function. c. Algebraic Write a quadratic equation in standard form with 2 + i and 2 i as its roots. d. Graphical Graph the quadratic equation found in part c by graphing its related function. e. Analytical How do you know when a quadratic equation will have only complex solutions? a. x = 0 b. c. x 2 4x + 5 = 0 d. e. A quadratic equation will have only complex solutions when the graph of the related function has no x-intercepts. esolutions Manual - Powered by Cognero Page 17
18 66. ERROR ANALYSIS Joe and Sue are simplifying (2i)(3i)(4i). Is either of them correct? Explain your reasoning. Sue; i 3 = i, not CHALLENGE Simplify (1 + 2i) REASONING Determine whether the following statement is always, sometimes, or never true. Explain your reasoning. Every complex number has both a real part and an imaginary part. Always. The value of 5 can be represented by 5 + 0i, and the value of 3i can be represented by 0 + 3i. 69. OPEN-ENDED Write two complex numbers with a product of 20. (4 + 2i)(4 2i) = 16 8i + 8i + 4 = = WRITING IN MATH Explain how complex numbers are related to quadratic equations. Some quadratic equations have complex solutions and cannot be solved using only the real numbers. esolutions Manual - Powered by Cognero Page 18
19 71. Which expression is equivalent to 5 3( 7 + 4i) (8 i)? A i B i C 18 11i D i Choice C is correct. 72. The formula V = C I describes the relationship between voltage (V), current (C) and impedance (I). What is the voltage when the current in a circuit is 5 7j amps and the impedance is 4 + j ohms? A 13 18j volts B 13 23j volts C 27 18j volts D 27 23j volts The voltage is 27 23j volts, so D is the correct choice. esolutions Manual - Powered by Cognero Page 19
20 73. Which expression is equivalent to 1 i? A B C D E You know that the powers of i are cyclical:,,, Express and in terms of,,, or to simplify. Substitute and into each expression to determine which is equivalent to. The expression is equivalent to 1 i, so choice C is correct. esolutions Manual - Powered by Cognero Page 20
21 74. What are the values of x and y when (9 + 6i) (x + yi) = 2 + 4i? A x = 7, y = 2 B x = 7, y = 10 C x = 11, y = 2 D x = 11, y = 10 Set the real parts equal to each other. Set the imaginary parts equal to each other. The correct answer is C. 75. If a > 0 and (a 2i) (a + 2i) = 5 then what is the value of a? Because a > 0, the answer is a = MULTI-STEP Consider the expression. a. Which expressions are equivalent to the numerator? Choose all that apply. A B C D E F b. What is the complex conjugate of the denominator? c. Which expression is equivalent to the quotient? A B C D esolutions Manual - Powered by Cognero Page 21
22 a. Expressions B, D, and E are equivalent. b. c. Expression D is equivalent. esolutions Manual - Powered by Cognero Page 22
23 77. Which expressions are equal to? A B C D E F A, so A is equal. B C D E F, so B is not equal., so C is not equal., so D is equal., so E is not equal., so F is equal. esolutions Manual - Powered by Cognero Page 23
0-2 Operations with Complex Numbers
Simplify. 1. i 10 2. i 2 + i 8 3. i 3 + i 20 4. i 100 5. i 77 esolutions Manual - Powered by Cognero Page 1 6. i 4 + i 12 7. i 5 + i 9 8. i 18 Simplify. 9. (3 + 2i) + ( 4 + 6i) 10. (7 4i) + (2 3i) 11.
0-2 Operations with Complex Numbers
Simplify. 1. i 10 1 2. i 2 + i 8 0 3. i 3 + i 20 1 i esolutions Manual - Powered by Cognero Page 1 4. i 100 1 5. i 77 i 6. i 4 + i 12 2 7. i 5 + i 9 2i esolutions Manual - Powered by Cognero Page 2 8.
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2. Simplify each expression. 10. MULTIPLE CHOICE Which expression is equivalent to? A B 4. C D 6. 8. 12. The correct choice is D. Simplify each expression. esolutions Manual - Powered by Cognero Page 1
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